|
| |
|
|
A166310
|
|
Wythoff Triangle, T.
|
|
1
| |
|
|
1, 2, 3, 4, 6, 8, 5, 7, 9, 11, 10, 12, 14, 16, 21, 13, 15, 17, 19, 24, 29, 18, 20, 22, 27, 32, 37, 42, 23, 25, 30, 35, 40, 45, 50, 55, 26, 28, 33, 38, 43, 48, 53, 58, 63, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 34, 39, 44, 49, 54, 59, 64, 69, 74, 79, 84, 47, 52, 57, 62, 67, 72
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| (1) Every positive integer occurs exactly once, so that
this is a permutation of the natural numbers.
(2) Obtained from the preliminary Wyhoff triangle
(A166309) by arranging each row in increasing order.
(3) The difference between consecutive row terms is a
Fibonacci number (A000045).
(4) Is the difference between consecutive column terms a
Fibonacci number?
|
|
|
REFERENCES
| C. Kimberling, "The Wythoff triangle and unique representations of positive integers," Proceedings of the Fourteenth International Conference on Fibonacci Numbers and Their Applications," Aportaciones Matematicas Invertigacion 20 (2011) 155-169.
|
|
|
FORMULA
| For a=1,2,3,... and b=0,1,...,a-1, let P(a,b) be the
number of the row of the Wythoff array (A035513) that
precurses to (a,b). Then for each a, arrange the numbers P
(a,b) in increasing order.
|
|
|
EXAMPLE
| The first nine rows of T:
1
2....3
4....6...8
5....7...9..11
10..12..14..16..21
13..15..17..19..24..29
18..20..22..27..32..37..42
23..25..30..35..40..45..50..55
26..28..33..38..43..48..53..58..63
Row 5 of the preliminary Wythoff triangle is
16,21,10,12,14, so that row 5 of the Wythoff triangle is
10,12,14,16,21. These are the row numbers of the Wythoff
array W (A035513) which precurse to pairs (5,b) for
b=0,1,2,3,4, not respectively. Example of precursion: row
16 of W is 40,65,105,...; then 65-40=25, 40-25=15,
25-15=10, 15-10=5, 10-5=5, 5-5=0, 5-0=5, so that the
initial pair (5,0) is reached in seven precursive steps.
|
|
|
CROSSREFS
| Cf. A035513, A165357, A166309.
Sequence in context: A069912 A152306 A120817 * A109852 A083197 A038150
Adjacent sequences: A166307 A166308 A166309 * A166311 A166312 A166313
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), Oct 11 2009
|
| |
|
|
